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arxiv: 2604.23269 · v1 · submitted 2026-04-25 · 🧮 math.DS · math.OC

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WSINDy for Model Predictive Control with Applications to Fusion, Drones, and Chaos

Cristian L\'opez , Mckenna Partridge , Sebastian De Pascuale , Jeremy Lore , Andrew Christlieb , Stephen Becker , David M. Bortz
Authors on Pith no claims yet

Pith reviewed 2026-05-08 06:59 UTC · model grok-4.3

classification 🧮 math.DS math.OC
keywords WSINDymodel predictive controlsystem identificationdata-driven controlnoisy dataplasma boundary controldrone trackingLorenz system
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The pith

WSINDYc integrated with model predictive control identifies governing dynamics robustly from noisy data for improved control performance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper integrates WSINDy with actuation inputs into a model predictive control setup to create a data-driven controller. WSINDYc recovers the underlying equations and input effects from data even when noise is high. This results in controllers that track references better, avoid obstacles more reliably, and run with lower costs than other methods. The demonstrations cover plasma boundary control in a tokamak, drone flight with collision avoidance, the Lorenz attractor, and simplified aircraft dynamics. A reader would care because real control problems often lack clean models and must work with imperfect data.

Core claim

The paper establishes that embedding WSINDYc within MPC produces models that remain effective for control despite high noise levels in the data, yielding longer prediction horizons, lower tracking errors, more reliable obstacle clearance, and lower optimization costs across tokamak, drone, Lorenz, and aircraft examples.

What carries the argument

WSINDYc, which modifies the weak-form sparse identification method to include actuation input terms in the candidate function library for learning both the system and its response to controls.

If this is right

  • Longer prediction horizons become usable in MPC without loss of accuracy.
  • Trajectory tracking errors decrease compared to benchmark data-driven methods.
  • Obstacle clearance in drone scenarios becomes more reliable.
  • Overall MPC costs are lower when operating on noisy measurements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could apply to additional noisy control tasks such as autonomous driving or process control.
  • Sparsity in the learned models may enable faster computation for real-time use.
  • Testing with time-dependent or uncertain parameters would reveal further limitations or strengths.

Load-bearing premise

The model identified by WSINDYc from noisy data remains sufficiently accurate over the MPC prediction horizon for the chosen applications.

What would settle it

Increasing noise in the identification data until incorrect terms are selected by WSINDYc and checking if MPC performance then falls below that of alternative methods.

Figures

Figures reproduced from arXiv: 2604.23269 by Andrew Christlieb, Cristian L\'opez, David M. Bortz, Jeremy Lore, Mckenna Partridge, Sebastian De Pascuale, Stephen Becker.

Figure 1
Figure 1. Figure 1: The WSINDY–MPC framework. Offline stage: ➀ data generation and ➁ model identification using WSINDYc. Online stage: ➂ MPC implementation using the learned model. For example, the Lorenz 63 system, shown in plant, is driven toward a selected fixed point (marked as black dots) using MPC. Algorithm 1. WSINDY–MPC framework Input: System Identification (SI): Training data Y, U (sampled at ∆t), candidate library … view at source ↗
Figure 2
Figure 2. Figure 2: SOLPS-ITER configuration and mesh, adapted from [38], CC BY 4.0. Colored regions denote various plasma regions. (Online version in color.) on the reduced library (with 0.6 coefficient inclusion threshold for aggregation via median). We also employ (iv) dynamic mode decomposition with control (DMDc) [64], an extension of standard DMD [10], that includes actuation inputs and creates low-order linear models f… view at source ↗
Figure 3
Figure 3. Figure 3: Time series and models identified for the SOLPS-ITER are presented in the white region, while the MPC stage is represented in the pink region. The top rows are the states of interest, while the bottom row is the control input (Online version in color.) according to the coupled dynamics. Different from the implementation of [34], where each MPC prediction step advances the model by a single numerical integr… view at source ↗
Figure 4
Figure 4. Figure 4: Performance for different noise levels for the SOLPS-ITER example: (a) Average relative error between the obtained state and the reference signal, (b) Success rate. E-WSINDYc demonstrates better tracking of state x1 with generally lower relative error and higher success rates. Beyond the shaded region, performance degrades: models diverge and have no predictive power. (Online version in color.) () () Tr!"… view at source ↗
Figure 5
Figure 5. Figure 5: (a) Inertial (solid lines), body (dashed lines) reference frames, along with forces and moments acting on quadrotor, (b) Data generation using the PD controller (Online version in color.) offline stage, where a proportional-derivative (PD) controller is used to track reference trajectories chosen to sufficiently excite the quadrotor simulator, thus generating the state and input data required for system id… view at source ↗
Figure 6
Figure 6. Figure 6: (a) Trajectory tracking performance and obstacle avoidance, η = 0.05 and the obstacle radius is 10 cm; Performance for different noise levels: (b) mean square error, and (c) minimum obstacle clearance, for 200 noise realizations each. Outside the shaded region, prediction accuracy deteriorates, and the resulting control actions fail to guarantee safe obstacle avoidance. (Online version in color.) η = 0.01,… view at source ↗
Figure 7
Figure 7. Figure 7: Prediction and control results for the Lorenz attractor: (a) time series of states and inputs during training, validation, and control stage, (b) Cumulative cost of Eq. (3.1), and (c) Runtime of the MPC optimization process. (Online version in color.) t uw{|}~| € u|‚ € „ ƒƒ „ †ƒ‡ˆ‰ Š‹„ Œ  Ž  †ƒ‡ˆ‰ ‘ †ƒ‡ˆ‰ ‘ †ƒ‡ˆ‰ Time ’• 30 40 0.01 0.1 0.2 0.3 0.4 0.5 () ( ) 1 2 3  ™ š›šœ  ™ š›œ  ™ š›ž 10 1 1… view at source ↗
Figure 8
Figure 8. Figure 8: Validation prediction performance for different noise levels: (a) time series, showing median (thick colored line) and 25–75th percentiles (shaded region), and (b) prediction horizon in time units, from 200 noise realizations each (Online version in color.) 5; outliers are removed using MATLAB’s rmoutliers function, and the remaining data are smoothed with smoothdata using a Gaussian window of length 10. W… view at source ↗
Figure 9
Figure 9. Figure 9: Validation prediction and control performance on the Lorenz system as the length of training data increases, when η = 0.1: (a) prediction horizon, (b) average relative prediction error, (c) training time, (d) terminal cumulative cost, using Eq. (3.1), over 5 time units, where from 800 data points onward, E-WSINDYc demonstrates more consistent control performance with generally lower costs, and (e) time ser… view at source ↗
Figure 10
Figure 10. Figure 10: Performance for different noise levels: MSE between the obtained state and the reference signal, from first to third quartile. E-WSINDYc generally shows less dispersion in the data. Beyond the shaded region, performance degrades, some models diverge, and have no predictive power. (Online version in color.) divertor conditions. The T div e,sep serves as a crucial indicator for the divertor integrity and sc… view at source ↗
Figure 11
Figure 11. Figure 11: Trajectory tracking performance and obstacle avoidance. Performance for different noise levels: (a) mean square error, and (b) minimum obstacle clearance, for 200 noise realizations each. Outside the shaded region, prediction accuracy deteriorates, and the resulting control actions fail to guarantee safe obstacle avoidance. (Online version in color.) controller [67] based on the public code of [78]. The r… view at source ↗
Figure 12
Figure 12. Figure 12: Lorenz System, first to third quartile: (a) Validation prediction performance for different noise levels, prediction horizon in time units, (b) different training data, terminal cumulative cost, using Eq. (3.1), over 5 time units, from 200 noise realizations each (Online version in color.) Appendix C. Lorenz system and simulation parameters The governing equations of this system are given by: x˙ 1 = 10(x2… view at source ↗
Figure 13
Figure 13. Figure 13: Prediction and control performance for the F-8 aircraft model: (a) time series of states and input during training: left panel for DMDc and SINDYc-based methods; right panel for NN, (b) Validation stage, (c) Reference (r) and angle of attack y = x1, (d) control input, and (e) cumulative cost over 6 time units during control. (Online version in color.) is ∆tM = 0.01 s and the plant’s ∆tplant = 0.001 s, and… view at source ↗
read the original abstract

The control of complex dynamical systems remains a fundamental challenge in science and engineering, where strong nonlinearities, the presence of noise, and computational constraints often pose significant obstacles in traditional control approaches. Recent advances in data-driven methods, particularly system identification techniques, have shown a powerful alternative by providing fast, parsimonious, interpretable models that are well-suited for model predictive control (MPC). Building on these developments, the present article embeds WSINDy with actuation inputs (WSINDYc) within a MPC framework. Compared to benchmark data-driven methods, WSINDYc enables a more robust identification of the governing dynamics, particularly in the presence of high noise levels, resulting in more accurate and efficient control. The capabilities of the proposed WSINDY-MPC framework are demonstrated on a range of problems, including a tokamak plasma boundary model that includes main ion gas puff actuation, drone tracking and collision avoidance, the chaotic Lorenz system, and a simplified flight control model for an F-8 aircraft. The proposed framework achieves superior performance in the presence of noise, enabling longer prediction horizons, lower trajectory tracking error, and a more reliable obstacle clearance, while simultaneously achieving lower MPC cost values compared to the baseline methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript introduces WSINDYc (WSINDy with actuation inputs) and embeds it in an MPC framework for data-driven control of nonlinear systems. It claims that WSINDYc yields more robust identification of governing equations from noisy measurements than benchmark methods, enabling longer prediction horizons, lower tracking errors, better obstacle clearance, and reduced MPC costs. These claims are illustrated on a tokamak plasma boundary model with gas-puff actuation, drone tracking/collision avoidance, the chaotic Lorenz system, and an F-8 aircraft flight model.

Significance. If the central performance claims are supported by explicit verification that the identified models maintain bounded multi-step prediction error over the MPC horizon, the work would offer a practical, parsimonious route to robust data-driven MPC for noisy nonlinear systems. The breadth of the four test cases (including a fusion-relevant tokamak model and chaotic dynamics) would strengthen the evidence for applicability beyond single benchmarks.

major comments (1)
  1. The central claim requires that WSINDYc-identified models remain sufficiently accurate over the MPC prediction horizon under noise. The applications demonstrate lower closed-loop tracking error and cost, but the manuscript does not report separate open-loop multi-step prediction error metrics (e.g., normalized RMSE over the horizon length) for the WSINDYc models versus baselines at the noise levels used. In the Lorenz and tokamak sections, this separation is needed to confirm that gains originate from identification robustness rather than cost-function tuning or constraint handling, as small coefficient perturbations can produce exponential divergence in these regimes.
minor comments (3)
  1. Abstract: the statements of 'superior performance' and 'more accurate and efficient control' should be accompanied by at least one quantitative summary statistic (e.g., percentage reduction in tracking error or extension of feasible horizon) with reference to the relevant figure or table.
  2. Ensure all comparison plots include explicit noise-level annotations, error bars or shaded regions indicating variability across trials, and clear specification of the baseline methods (e.g., which SINDy variant or neural-network identifier is used).
  3. Provide a brief sensitivity analysis or table showing how WSINDYc hyper-parameters (library size, threshold, regularization) affect both identification accuracy and closed-loop MPC cost for at least one application.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for major revision. The central concern is well-taken: while closed-loop metrics are the ultimate test of the WSINDYc-MPC framework, separate open-loop multi-step prediction errors would more clearly isolate the contribution of improved identification robustness. We will address this directly in the revision.

read point-by-point responses

  1. Referee: The central claim requires that WSINDYc-identified models remain sufficiently accurate over the MPC prediction horizon under noise. The applications demonstrate lower closed-loop tracking error and cost, but the manuscript does not report separate open-loop multi-step prediction error metrics (e.g., normalized RMSE over the horizon length) for the WSINDYc models versus baselines at the noise levels used. In the Lorenz and tokamak sections, this separation is needed to confirm that gains originate from identification robustness rather than cost-function tuning or constraint handling, as small coefficient perturbations can produce exponential divergence in these regimes.

    Authors: We agree that explicit open-loop multi-step prediction metrics would strengthen the manuscript. Although the closed-loop results (tracking error, cost, obstacle clearance) already demonstrate practical utility, they do not fully separate model quality from controller tuning. In the revised manuscript we will add normalized RMSE (or equivalent) for open-loop multi-step predictions over the exact MPC horizons used, evaluated at the noise levels reported in each example. These will be included for WSINDYc and all baselines in the Lorenz and tokamak sections (and, space permitting, the drone and aircraft cases). We will also note that the identified models keep these errors bounded over the horizon, consistent with the longer prediction horizons that become feasible. This addition directly addresses the referee’s request without changing the core claims or experimental setup. revision: yes

Circularity Check

0 steps flagged

No significant circularity; WSINDy-MPC is an applied framework with independent empirical validation

full rationale

The paper presents WSINDYc as an existing data-driven identification technique (from prior literature) embedded into an MPC loop, then demonstrates closed-loop performance on four distinct applications (tokamak, drone, Lorenz, F-8). No derivation step equates a claimed prediction or uniqueness result to a fitted quantity defined by the same equations; the central claims rest on numerical experiments comparing tracking error, cost, and robustness under noise, which are externally falsifiable against the benchmark methods. Self-citations to the WSINDy method itself are not load-bearing for the MPC performance results, as the identification step uses standard sparse regression on observed trajectories and the MPC uses the resulting model as a black-box predictor. The weakest assumption (model fidelity over the horizon) is acknowledged as an empirical question rather than a definitional identity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are stated; the framework inherits standard assumptions of SINDy-type methods and MPC.

pith-pipeline@v0.9.0 · 5537 in / 1058 out tokens · 38502 ms · 2026-05-08T06:59:55.998036+00:00 · methodology

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